Good reduction of periodic points on projective varieties

Hutz, Benjamin

doi:10.1215/ijm/1290435342

Cited by 11 publications

(26 citation statements)

References 10 publications

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“…The following theorem is an amalgamation of results. We have stated it for maps of P N defined over Q, but it holds much more generally for self-maps of algebraic varieties over number fields; see especially [113]. If one restricts to polynomial maps on P 1 , then Benedetto [31] proves that one can take C poly (1, d, D, s) equal to a mutiple of d 3 (D + s) log(D + s), while Canci and Paladino [48] give a weaker bound for rational maps on P 1 .…”

Section: Good Reduction Of Maps and Orbitsmentioning

confidence: 99%

Current trends and open problems in arithmetic dynamics

Benedetto¹,

Ingram²,

Jones³

et al. 2019

Bull. Amer. Math. Soc.

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Arithmetic dynamics is the study of number theoretic properties of dynamical systems. A relatively new field, it draws inspiration partly from dynamical analogues of theorems and conjectures in classical arithmetic geometry, and partly from p-adic analogues of theorems and conjectures in classical complex dynamics. In this article we survey some of the motivating problems and some of the recent progress in the field of arithmetic dynamics. 1 2 BENEDETTO, ET AL. 22. Local-Global Questions in Dynamics 61 References 63 TRENDS AND PROBLEMS IN ARITHMETIC DYNAMICS 3background on dynamical systems, number theory, and algebraic geometry that we will need. As noted at the end of Section 3, we have attempted to make this article accessible by generally restricting attention to maps of projective space, thereby obviating the need for more advanced material from algebraic geometry. Our survey of arithmetic dynamics then commences in Section 4 and concludes in Section 22. These nineteen sections are mostly independent of one another, albeit liberally sprinkled with instructive cross-references.A Brief Timeline of the Early Days. The study of Galois groups of iterated polynomials was initiated by Odoni in a series of papers [177,178,179] during the mid-1980s. Beyond this, it appears that the first paper to describe dynamical analogues for a wide range of classical problems in arithmetic geometry did so not for polynomial maps or projective space, but rather for K3 surfaces admitting an automorphism of infinite order. The 1991 paper of Silverman [204] discusses dynamical analogues of various problems, including: (1) Uniform boundedness of periodic points;(2) Periodic points on subvarieties;(3) Lower bounds for canonical heights; (4) Open image of Galois groups; (5) Integral points in orbits. The rest of the 1990s saw an explosion of papers in which numerous researchers began to study a wide variety of problems in arithmetic dynamics. The serious study of p-adic dynamics starts with a 1983 paper of Herman and Yoccoz [107], two of the world's leading complex dynamicists, in which they investigate a p-adic analogue of a classical problem in complex dynamics. Later in the 1980s and 90s, the physics literature includes several papers [6,7,21,230] that discuss p-adic dynamics for polynomial maps, but the deeper study of p-adic dynamics really took off with the Ph.D. theses of Benedetto [24, 25] in 1998 and Rivera-Letelier [190, 191] in 2000. The dynamics of iterated p-adic power series was investigated by Lubin [156] in a 1994 paper. We also mention that there is an extensive literature on ergodic theory over p-adic fields which, although not within the purview of this survey, dates back to at least 1975 [180]. Abstract Dynamical SystemsA discrete dynamical system consists of a set S and a self-map

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Section: Good Reduction Of Maps and Orbitsmentioning

confidence: 99%

Current trends and open problems in arithmetic dynamics

Benedetto¹,

Ingram²,

Jones³

et al. 2019

Bull. Amer. Math. Soc.

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“…Therefore, the length of the orbit of Y is the length of the orbit of α i (under the action of f i ). Because α i ∈ P 1 K is a periodic point for f i and moreover, f i has good reduction modulo p, then the length of the orbit of α i is bounded only in terms of p, e, [κ : F p ], as proven in [Zie96,Hut09]; hence the length of the orbit of Y is bounded by a constant which is independent of Y .…”

Section: The Case Of Curvesmentioning

confidence: 94%

“…In [MS94], Morton and Silverman conjecture that there is a constant C(N, d, D) such that for any morphism f : P N −→ P N of degree d defined over a number field K with [K : Q] ≤ D, the number of preperiodic points of f over K is less than or equal to C(N, d, D). This conjecture remains very much open, but in the case where f has good reduction at a prime p, a great deal has been proved about bounds depending on p, N , d, D (see [Zie96,Pez05,Hut09]).…”

Section: Introductionmentioning

confidence: 99%

Bounding periods of subvarieties of $(\mathbb{P}^1)^n$

Bell

Ghioca

Tucker

2019

Mathematical Research Letters

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“…Let f : P N → P N be a morphism of degree d defined over K. Let X ⊂ P N defined over K be an irreducible periodic subvariety of degree D and codimension t for f with minimal period n. Let p ∈ K be a prime of good reduction for f . If deg(f ℓ (X)) = deg(f ℓ (X)) for 0 ≤ ℓ ≤ n, then there exists a constant C depending only on d, D, N, t, p such that n ≤ C (d, D, N, t, p).The basic idea of the method is to move the problem to an endomorphism of a component of the Chow variety and appeal to the similar bound for points from the author's previous work [16]. The restriction on the degree can be thought of as primes of good reduction for the subvariety and can only not be satisfied for finitely many primes for a given periodic subvariety.…”

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confidence: 99%

Good reduction and canonical heights of subvarieties

Hutz

2018

Mathematical Research Letters

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We bound the length of the periodic part of the orbit of a preperiodic rational subvariety via good reduction information. This bound depends only on the degree of the map, the degree of the subvariety, the dimension of the projective space, the degree of the number field, and the prime of good reduction. As part of the proof, we extend the corresponding good reduction bound for points proven by the author for non-singular varieties to all projective varieties. Toward proving an absolute bound on the period for a given map, we the bound between the height and canonical height of a subvariety via Chow forms. This gives the existence of a bound on the number of preperiodic rational subvarieties of bounded degree for a given map. An explicit bound is given for hypersurfaces.Theorem. (Theorem 3.8) Let K be a number field. Let f : P N → P N be a morphism of degree d defined over K. Let X ⊂ P N defined over K be an irreducible periodic subvariety of degree D and codimension t for f with minimal period n. Let p ∈ K be a prime of good reduction for f . If deg(f ℓ (X)) = deg(f ℓ (X)) for 0 ≤ ℓ ≤ n, then there exists a constant C depending only on d, D, N, t, p such that n ≤ C (d, D, N, t, p).The basic idea of the method is to move the problem to an endomorphism of a component of the Chow variety and appeal to the similar bound for points from the author's previous work [16]. The restriction on the degree can be thought of as primes of good reduction for the subvariety and can only not be satisfied for finitely many primes for a given periodic subvariety. However, this is probably not quite enough for

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Good reduction of periodic points on projective varieties

Cited by 11 publications

References 10 publications

Current trends and open problems in arithmetic dynamics

Current trends and open problems in arithmetic dynamics

Bounding periods of subvarieties of $(\mathbb{P}^1)^n$

Good reduction and canonical heights of subvarieties

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